Measurable n -sensitivity and maximal pattern entropy

We introduce the notion of measurable n -sensitivity for measure preserving systems, and study the relation between measurable n -sensitivity and the maximal pattern entropy. We prove that, if ( X , ℬ , μ , T ) is ergodic, then ( X , ℬ, μ , T ) is measurable n -sensitive but not measurable ( n + 1)-sensitive if and only if h * μ ( T ) = log n , where h* μ ( T ) is the maximal pattern entropy of T .